2,823 research outputs found
On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations
We construct Miura transformations mapping the scalar spectral problems of
the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS)
list into the discrete Schr\"odinger spectral problem associated with
Volterra-type equations. We show that the ABS equations correspond to
B\"acklund transformations for some particular cases of the discrete
Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to
construct new generalized symmetries for the ABS equations. The same can be
said about the generalizations of the ABS equations introduced by Tongas,
Tsoubelis and Xenitidis. All of them generate B\"acklund transformations for
the YdKN equation. The higher order generalized symmetries we construct in the
present paper confirm their integrability.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Poisson integrators
An overview of Hamiltonian systems with noncanonical Poisson structures is
given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are
presented. Numerical integrators using generating functions, Hamiltonian
splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems
and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems
and the discrete gradient methods are also presented.Comment: 30 page
Lie point symmetries of differential--difference equations
We present an algorithm for determining the Lie point symmetries of
differential equations on fixed non transforming lattices, i.e. equations
involving both continuous and discrete independent variables. The symmetries of
a specific integrable discretization of the Krichever-Novikov equation, the
Toda lattice and Toda field theory are presented as examples of the general
method.Comment: 17 pages, 1 figur
Non-standard discretization of biological models
We consider certain types of discretization schemes for differential equations with quadratic nonlinearities, which were introduced by Kahan, and considered in a broader setting by Mickens. These methods have the property that they preserve important structural features of the original systems, such as the behaviour of solutions near to fixed points, and also, where appropriate (e.g. for certain mechanical systems), the property of being volume-preserving, or preserving a symplectic/Poisson structure. Here we focus on the application of Kahan's method to models of biological systems, in particular to reaction kinetics governed by the Law of Mass Action, and present a general approach to birational discretization, which is applied to population dynamics of Lotka-Volterra type
Einstein-Cartan theory as a theory of defects in space-time
The Einstein-Cartan theory of gravitation and the classical theory of defects
in an elastic medium are presented and compared. The former is an extension of
general relativity and refers to four-dimensional space-time, while we
introduce the latter as a description of the equilibrium state of a
three-dimensional continuum. Despite these important differences, an analogy is
built on their common geometrical foundations, and it is shown that a
space-time with curvature and torsion can be considered as a state of a
four-dimensional continuum containing defects. This formal analogy is useful
for illustrating the geometrical concept of torsion by applying it to concrete
physical problems. Moreover, the presentation of these theories using a common
geometrical basis allows a deeper understanding of their foundations.Comment: 18 pages, 7 EPS figures, RevTeX4, to appear in the American Journal
of Physics, revised version with typos correcte
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